Congruences and ramified primes in fields of coefficients of newforms

Abstract: We investigate the splitting behavior of $\ell$ in the coefficient field of a newform $f$ of level $N$, under the assumption that $f$ is congruent modulo a prime above $\ell$ to another newform $g$ whose level divides $N/p^2$ for some prime $p\mid N$. In particular, we show that the maximal real subfield of the $\ell$-th cyclotomic field, $\mathbb{Q}(ζ\ell + ζ\ell^{-1})$, is contained in the coefficient field of $f$. We conclude by presenting explicit examples that illustrate these results.

• The paper establishes that congruences between newforms force the inclusion of cyclotomic subfields within coefficient fields.
• It employs detailed local and global analyses of Galois representations through level-raising techniques and explicit computational verifications.
• The results generalize previous findings by providing new lower bounds for ramification indices and structural constraints on modular form coefficients.

Ramification and Cyclotomic Subfields in Coefficient Fields of Newforms

Introduction and Motivation

The paper "Congruences and ramified primes in fields of coefficients of newforms" (2603.29468) systematically studies the ramification and splitting behavior of primes in the coefficient fields of classical and Hilbert newforms, especially under congruence relations with other newforms of related but lower level. The central focus is the interplay between local ramification at a prime $p \mid N$ in the modular form's level and the arithmetic of the coefficient field, particularly for primes $\ell$ appearing in congruences between newforms.

By leveraging explicit analyses of associated Galois representations, the authors establish strong structural constraints on the inclusion of cyclotomic subfields in the coefficient fields, as well as lower bounds for ramification indices at primes above $\ell$. These results generalize earlier conclusions of Brumer and others by combining level-raising and congruence arguments.

Main Results

The principal theorem asserts the following: Let $f$ be a newform of weight $k \geq 2$ and level $N$, and suppose $p^2 \mid N$. For an odd prime $\ell \neq p$, if there exists a congruence modulo a prime above $\ell$ with another newform $g$ of level dividing $\ell$0, and the reduction of $\ell$1 modulo $\ell$2 loses ramification at $\ell$3, then the coefficient field $\ell$4 contains a subfield directly related to the $\ell$5-th cyclotomic field.

Specifically:

• When $\ell$6 and ramification at $\ell$7 is completely lost in the mod $\ell$8 representation, the maximal real subfield $\ell$9 is contained in $\ell$0, and any prime over $\ell$1 has ramification degree divisible by $\ell$2.
• If the mod $\ell$3 reduction is irreducible, the ramification degree is at least $\ell$4 for one prime above $\ell$5.
• For $\ell$6, the full cyclotomic field $\ell$7 embeds into $\ell$8.

These claims extend previous work by removing restrictions on the nebentypus and imposing stronger field containment when more ramification is lost. The proofs utilize the classification of possible inertial types at $\ell$9 for the relevant Galois representations, particularly Carayol's results on local types under conductor drop.

Methods

The approach combines:

• Detailed local and global analysis of the Galois representations $f$0 attached to $f$1, including reductions and inertial types.
• Carayol's classification of representations that lose ramification upon reduction, distinguishing principal series, special (Steinberg), and supercuspidal types.
• Cyclotomic character evaluation via local class field theory to demonstrate field inclusions.
• Explicit computation of ramification indices in the context of torsion-free conditions for certain pro-$f$2 groups (Silverberg-Zarhin lemma).
• Leveraging level-raising theorems (Diamond-Taylor) to systematically construct examples and verify theoretical claims computationally.

A critical aspect is the use of congruences modulo $f$3 that facilitate ramification loss and link newforms across levels, thereby influencing the arithmetic of their coefficient fields.

Explicit Examples and Computational Evidence

The paper presents numerous explicit cases, often computed using the LMFDB database, to illustrate the predicted field inclusions and ramification indices. For selected pairs $f$4 and newforms $f$5 congruent to $f$6, the authors detail the structure and degrees of the coefficient fields $f$7, confirming divisibility of ramification degrees by $f$8, and in some cases, achieving equality.

A systematic computational strategy based on the Diamond-Taylor level-raising theorem is described: start from a low-level newform $f$9 with irreducible mod $k \geq 2$0 representation, find a suitable prime $k \geq 2$1 satisfying level-raising conditions, and construct a newform $k \geq 2$2 at level $k \geq 2$3 congruent to $k \geq 2$4 mod $k \geq 2$5, then analyze splitting of $k \geq 2$6 in $k \geq 2$7. The results match theoretical expectations and highlight practical methods for generating newforms with controlled ramification in their coefficient fields.

Theoretical and Practical Implications

This work advances the understanding of how local ramification and level structure in modular forms dictate the arithmetic of their coefficient fields, especially in the context of congruence relations. The consequences are twofold:

• Theoretical: The results provide new instances where the cyclotomic fields and their real subfields are forced into modular coefficient fields, offering additional constraints useful in the study of the inverse Galois problem and modularity lifting. The ramification behavior clarifies finer aspects of the relationship between modular forms and their associated Galois representations.
• Practical: The explicit computational methods open avenues for constructing modular forms with prescribed ramification properties, which may be used to probe modularity questions, test inverse Galois realizations, or design forms with desired local-global behavior.

The approach generalizes well to Hilbert modular forms of arithmetic weight, suggesting broader applicability in automorphic settings.

Outlook and Future Directions

Future developments may include:

• Expansion to higher powers of $k \geq 2$8 in the level, albeit at the cost of substantially larger coefficient fields and computational complexity.
• Investigation of similar phenomena for other types of automorphic forms and representations.
• Utilization in explicit inverse Galois constructions where ramification control is crucial.
• Deepening the connection between congruence phenomena and the structure of the Hecke algebra for various levels and weights.

The combination of theoretical classification, congruence relations, and computational verification sets a robust foundation for further exploration of ramification in modular and automorphic settings.

Conclusion

This study provides rigorous results on the containment of cyclotomic subfields within coefficient fields of newforms, arising from congruences and local ramification loss. The precise bounds and explicit computational evidence reinforce a strong link between local representation theory and the arithmetic of modular forms, contributing meaningful constraints to the field of modular forms and Galois representations (2603.29468).